In the present paper we calculate the 5PN \(O(\nu ^2)\)  terms and perform a series of comparisons to the literature.
In Section  we calculate the 5PN local tail terms. To compare with Ref. [1]} we express
our results for the local contributions obtained in the harmonic gauge in terms of effective one body (EOB)
potentials in Section . To fix the five new EOB parameters at 5PN we are solely using
the local contributions to
periastron advance
\(K^{\rm loc,h}(\hat{E},j)\) , which is an observable [2]} since the non–local terms are known
in explicit form via an eccentricity expansion, cf. [1]}, [4]}, and does not require any
regularization.Regularizations may imply scheme dependencies. Furthermore, we will not use assumptions
on the \(\nu \)  dependence of the observables used but perform a direct calculation within the framework of an
effective field theory approach. The binding energy and periastron advance in the circular case are used for
consistency checks for two parameters. Finally, we also discuss the determination of the 5PN parameters using
the expansion coefficients of the scattering angle and compare to the literature. We summarize phenomenological
results in Section  and Section  contains the conclusions. Some technical
aspects are given in the Appendices – on the Feynman rules, invariant functions, the
in–in formalism, calculation of the tail terms and relations for the scattering angle.
